Integer-DigitFunctions: AnExample of Math-Art IntegrationERNESTO ESTRADA

‘‘God made the integers, all else is the work of man.’’L. Kronecker

MMathematics and the visual arts mutually reinforceone another [10]. On the one hand, many math-ematical objects appear in artistic or decorative

works [5, 6]. In particular, mathematical curves and art havea long-standing connection through the application ofgeometric principles [13, 20]. Simple curves such as thecatenary are ubiquitous, for example in the work of theCatalan architect Antoni Gaudı, as well as in ancient [19]and in modern architecture [12]. On the other hand, manymathematical objects display artistic appeal per se. The vastlist includes, among others: knots [1, 2], mosaics and tiles[3, 21], Fourier series [9], topological tori [15], and fractalcurves [4], all of which produce visual patterns of undeni-able beauty. Such connections between mathematics andthe arts are explored and celebrated annually by theBridges community formed by mathematicians and artists[8, 18].

This article introduces a new family of curves in theplane defined by functions transforming an integeraccording to sums of digit-functions. These transforms ofan integer N are obtained by multiplying the function valuef Nð Þ by the sum of f aið Þ, where ai are the digits of N in agiven base b and f is a standard function such as atrigonometric, logarithmic, or exponential one. Thesecurves display a few attributes, such as beauty, symmetry,and resemblance to natural environments, which makethem attractive for artistic purposes.

Digit Sum with ‘‘Memory’’A nonnegative integer N can be represented in a given baseb according to the following expression

N ¼ a1bn þ a2b

n�1 þ a3bn�2 þ � � � þ an�1bþ an; ð1Þ

where ai are nonnegative integers. We can represent anegative integer in a similar way if we multiply by �1 theright-hand-side part of the previous equation. For instance,the number N ¼ 257, expressed in the base b ¼ 10, is:

257 ¼ 2 � 102 þ 5 � 10 þ 7:

A function explored in several contexts [14] is the ‘‘digitsum,’’ or ‘‘sum of digits’’ of a given integer, defined by:

Sb Nð Þ ¼Xn

i¼1

ai ¼Xlogb Nb c

k¼0

1

bkNmodbkþ1 � Nmodbk� �

: ð2Þ

For instance, for b ¼ 10 the digit sum of 257 is2 þ 5 þ 7 ¼ 14. The digit sum forms an integer sequenceregistered in the Online Encyclopedia of Integer Sequences(OEIS) [16, 17] by the sequence A007953. One characteristicof the digit sum function is that it does not ‘‘remember’’ theinteger that gave rise to it. That is, we can find many dif-ferent integers for which the digit sum is the same. Forinstance, S10 59ð Þ ¼ S10 77ð Þ ¼ S10 95ð Þ ¼ S10 149ð Þ ¼ � � � ¼S10 257ð Þ ¼ � � � ¼ 14.

� 2018 The Author(s). This article is an open access publication, Volume 40, Number 1, 2018 73

https://doi.org/10.1007/s00283-017-9726-x

A simple modification of the digit sum function that‘‘remembers’’ its origins better can be obtained by multi-plying the digit sum of N by N [7]. Hereafter we represent

this function by bN . In the previous example we haved257 ¼ 257 � 14 ¼ 3598. This function is not different forevery integer, thus we cannot say that it remembers itsorigins perfectly, but at least it is not as degenerate as thedigit sum. The corresponding integer sequence is given atthe OEIS by A117570 [16]. In Figure 1 we illustrate the plot

of bN versus N for the first 1000 integers.

Integer-Digit FunctionsNow we extend this idea to other functions. Let f Nð Þbe a function of the integer N, for example, sin Nð Þ.Then we define the sum of digit-functions df Nð Þ of f Nð Þ by:

df Nð Þ ¼ f a1ð Þ þ f a2ð Þ þ � � � þ f anð Þð Þf Nð Þ: ð3Þ

Here, once again, we are using the previous notation; thatis, the nonnegative integers ai are the digits of N in base b.

For instance, dsin 257ð Þ¼ sin 2ð Þþsin 5ð Þþsin 7ð Þð Þ�sin 257ð Þ��0:2357. For negative integers �N , if the function f �Nð Þexists, we obtain

df �Nð Þ ¼ f a1ð Þþf a2ð Þþ���þf anð Þð Þf �Nð Þ: ð4Þ

As an example, the plot of dsin Nð Þ versus N is given in Figure 2(left panel), which mildly resembles the original sine function.

Our main interest here is, however, to make a com-plete transformation of both coordinate axes in terms ofour digit-function. That is, we are interested in the

plotting of df Nð Þ versus dg Nð Þ. For instance, when dsin Nð Þis plotted versus bN in Figure 2 (right panel), a totallyunexpected pattern emerges. This pattern does notresemble at all the original function and, if rotated by�90 degrees, it resembles a bush.

In the rest of this article we will be interested in the curves

we obtain from certain combinations of df Nð Þ and dg Nð Þ. Torepresent the curves we use here the following method. Fora given even number of points n, the curve we want is

obtained for the integers N � n

2using a parametric form of

the equation,

x ¼ df Nð Þ ; ð5Þ

y ¼ dg Nð Þ: ð6Þ

0 200 400 600 800 1000x

0

0.5

1

1.5

2

2.5

3x

×104

Figure 1. Plot of bN versus N for integers less than or equal to

1000.

0 200 400 600 800 1000x

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

sin(x)

0 0.5 1 1.5 2 2.5 3x ×104

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

sin(x)

Figure 2. Semideconstruction of the sine function (left) and total deconstruction of the same (right).

74 THE MATHEMATICAL INTELLIGENCER

If the functions df Nð Þ and dg Nð Þ are also defined for nega-tive arguments, the corresponding transforms are obtained

for �n

2�N .

All trigonometric functions are obtained in degrees. Forinstance, let

x ¼ bN ; ð7Þ

y ¼ dsinN � bN � dcosN : ð8Þ

Then for n ¼ 1000, we obtain the curve illustrated in theleft panel of Figure 3. Notice that the curves are discretebecause we only evaluate our functions at integer values.However, we can join the points to create a more fluidform; see the right panel of Figure 3, in which we used thecolormap function available in Matlab�.1

Artistic Creation Using Integer-Digit FunctionsNow we play with this construction and transform somewell-known curves, such as sine, involute of a circle,astroid, and others. Let us start by the simple sine functionin which we make the transformation

x ¼ bN ; ð9Þ

y ¼ dsinN : ð10Þ

This function is illustrated in Figure 4 for n ¼ 2000 points.The artistic representation of the curve given on the right-hand-side of the figure clearly resembles a butterfly and canserve as the basis for more artistic inspirations.

We now explore the transformation of both coordinateaxes for the involute of a circle. This function is given bythe following parametric equation

x ¼ dsinN � bN � dcosN ; ð11Þ

y ¼ dcosN þ bN � dsinN : ð12Þ

and the corresponding curve is illustrated in Figure 5.The analogue of the astroid using the integer-digit

transform is given by

x ¼ dcosN� �3

; ð13Þ

y ¼ dsinN� �3

; ð14Þ

and the corresponding curve for n ¼ 10; 000 points isillustrated in Figure 6.

The cardioid curve can be transformed using the fol-lowing parametric equations

x ¼ 2 dcosN þ dcos 2Nð Þ; ð15Þ

y ¼ 2 dsinN þ 3 dsin 2Nð Þ: ð16Þ

Figure 3. Curve defined by Equations (7)–(8) with n ¼ 1000

points. The left panel illustrates the discrete character of the

curve. Note also that negative inputs are distinguishable from

positive inputs. In the image on the left, red is used for

negative values of N and blue for positive ones. The image on

the right is obtained by joining the dots with lines to give a

more fluid view. In the right panel we used a ‘‘Jet’’ colormap to

color the curve.

Figure 4. Sine after the integer-digit transform using n ¼ 2000

points (left) and another view of it (right) using colormap ‘‘Jet’’

for artistic impression. To prepare the second image, we

simply rotate the negative part of the curve to obtain a mirror

symmetry. On the left-hand side we use red for negative values

of N and blue for positive ones.

Figure 5. Involute of the circle after the integer-digit trans-

form using n ¼ 4000 points (left) and a coloring of it using

‘‘Parula’’ color map (right).

1All images in this article were created in Matlab�. Any interested reader can obtain the code from the author via e-mail.

� 2018 The Author(s). This article is an open access publication, Volume 40, Number 1, 2018 75

In Figure 7 we plot the curve corresponding to the trans-formed cardioid, which resembles a fish.

Although ‘‘beauty is in the eye of the beholder’’ there isno doubt that these plots have certain artistic attributes thatoverall make them in some way unexpectedly beautiful. Inthe next section we explore some further possibilities forartistic creation.

Artistic CompositionsLet us start by seeing what we get when we composefunctions. We combine the function:

x ¼ dsinN � bN � dcosN ; ð17Þ

y ¼ bN ; ð18Þ

with

x ¼ 3000 dsinN � 1000; ð19Þ

y ¼ bN ; ð20Þ

and

x ¼ �3000 dsinN þ 1000; ð21Þy ¼ bN ; ð22Þ

to make the composition illustrated in the Figure 8.Next we introduce logarithmic functions into the mix.

The function

Figure 6. Astroid after the integer-digit transform using n ¼10; 000 points (left) and a coloring of it using the colormap

‘‘Hot’’ and a grey background (right).

Figure 7. Cardioid after the integer-digit transform using n ¼50; 000 points and using a combination of ‘‘Jet’’ and ‘‘Cooper’’

colormaps.

Figure 8. Composition of integer-digit transform (left) and

another view of the same composition (right).

Figure 9. Curve obtained by using logarithmic integer-digit transform for n ¼ 5000 points (left) and two modified views of it. Red

is used for negative values of t and blue for positive ones.

76 THE MATHEMATICAL INTELLIGENCER

x ¼ dsinN ; ð23Þ

y ¼ dlnN ; ð24Þ

is illustrated in the left-hand-side of Figure 9. The secondview was created with some of the filters available inPowerPoint. Here, as the log function is not defined for

negative numbers we plot � dsinN versus dlnN , and dsinN

versus dlnN to produce a symmetric image. A very simplemodification in the last image produces what in the eyes ofthe beholder resembles the face of a fox.

Finally in Figure 10, we present two naive combinationsof some of the previously obtained integer-digit transforms

to produce a landscape and a seascape. In the top panel weshow the combination of functions used for producingFigures 4 and 8. In the bottom panel we use the function

x ¼ bN ; ð25Þ

y ¼ dlnN ; ð26Þ

to produce the plants and then combine it with the functionused to produce Figure 7 to make the final composition.

The artistic compositions presented here may seemsimple and naive. But is it not surprising that they wereentirely produced by a mathematical procedure based onlyon integers– with just a little extra imagination?

ACKNOWLEDGMENTS

The author thanks artist Puri Pereira for useful discussions.

He also thanks the Royal Society of London for a Wolfson

Research Merit Award. Finally he thanks Gizem Karaali for

her editorial assistance.

Department of Mathematics & Statistics

University of Strathclyde

Livingstone Tower, 26 Richmond Street

Glasgow G1 1XQ

UK

e-mail: ernesto.estrada@strath.ac.uk

OPEN ACCESS

This article is distributed under the terms of the Creative

Commons Attribution 4.0 International License (http://

creativecommons.org/licenses/by/4.0/), which permits

unrestricted use, distribution, and reproduction in any

medium, provided you give appropriate credit to the

original author(s) and the source, provide a link to the

Creative Commons license, and indicate if changes were

made.

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